Randall D. Knight — Oscillations: Problem 58

It has recently become possible to “weigh” DNA molecules by measuring the influence of their mass on a nano-oscillator. FIGURE P15.58 shows a thin rectangular cantilever etched out of silicon (density \(2300 \text{ kg/m}^3\)) with a small gold dot (not visible) at the end. If pulled down and released, the end of the cantilever vibrates with SHM, moving up and down like a diving board after a jump. When bathed with DNA molecules whose ends have been modified to bind with gold, one or more molecules may attach to the gold dot. The addition of their mass causes a very slight—but measurable—decrease in the oscillation frequency. A vibrating cantilever of mass \(M\) can be modeled as a block of mass \(\frac{1}{3}M\) attached to a spring. (The factor of \(\frac{1}{3}\) arises from the moment of inertia of a bar pivoted at one end.) Neither the mass nor the spring constant can be determined very accurately—perhaps to only two significant figures—but the oscillation frequency can be measured with very high precision simply by counting the oscillations. In one experiment, the cantilever was initially vibrating at exactly \(12 \text{ MHz}\). Attachment of a DNA molecule caused the frequency to decrease by \(50 \text{ Hz}\). What was the mass of the DNA?